Question

\[\frac{dy}{dx} = x^5 + x^2 - \frac{2}{x}, x \neq 0\]

Answer 1

We have,
\[\frac{dy}{dx} = x^5 + x^2 - \frac{2}{x}\]
\[ \Rightarrow dy = \left( x^5 + x^2 - \frac{2}{x} \right)dx\]
Integrating both sides, we get
\[ \Rightarrow \int dy = \int\left( x^5 + x^2 - \frac{2}{x} \right)dx\]
\[ \Rightarrow y = \frac{x^6}{6} + \frac{x^3}{3} - 2\log\left| x \right| + C\]
\[\text{Clearly, } y = \frac{x^6}{6} + \frac{x^3}{3} - 2\log\left| x \right| + \text{ C is defined for all }x \in R \text{ except }x = 0 . \]
\[\text{ Hence, }y = \frac{x^6}{6} + \frac{x^3}{3} - 2\log\left| x \right| + \text{ C, where } x \in R - \left\{ 0 \right\}, \text{ is the solution to the given differential equation.}\]